How to use this calculator
Enter the three coefficients and the initial value and slope at t = 0. The calculator handles all three cases: distinct real roots, a repeated root and complex roots.
Method
Characteristic equationar² + br + c = 0
Distinct real rootsy = C1er₁t + C2er₂t
Repeated / complex rootsy = (C1 + C2t)ert y = eαt(C1 cos βt + C2 sin βt)
Worked example
y″ + 2y′ + 10y = 0, y(0) = 1, y′(0) = 0 (the default)
- r² + 2r + 10 = 0 gives Δ = 4 − 40 = −36, so r = −1 ± 3i.
- General solution: y = e−t(C1 cos 3t + C2 sin 3t).
- y(0) = C1 = 1; y′(0) = −C1 + 3C2 = 0, so C2 = 1/3.
- y = e−t(cos 3t + ⅓ sin 3t), and y(1) = −0.3469.
Frequently asked questions
What about a right-hand side (forcing)?
This calculator covers the homogeneous equation. For a forced equation, add a particular solution (undetermined coefficients or variation of parameters) to this homogeneous solution.
Where does this equation show up?
Spring-mass-damper systems, RLC circuits and pendulums (small angles) all reduce to it.